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Worked Examples · Example 19

Q.The supply function for a commodity is p=x2+4x+5p = x^2 + 4x + 5 where xx denotes supply. Find the producers' surplus when the price is 10.

Puducherry CbseNCERTSubjective· 3mImportance★★★★★
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At price p0=10p_0=10 supply is x0=1x_0=1; PS=p0x0−∫01(x2+4x+5) dx=83≈₹2.67PS=p_0x_0-\int_0^1 (x^2+4x+5)\,dx=\dfrac{8}{3}\approx ₹2.67.

Producer's surplus PS=p0x0−∫0x0g(x) dxPS=p_0x_0-\displaystyle\int_0^{x_0} g(x)\,dx, where p=g(x)p=g(x) is the supply curve, p0p_0 the market price and x0x_0 the quantity supplied at p0p_0.

  1. Supply: p=x2+4x+5p=x^2+4x+5. Set p=p0=10p=p_0=10: 10=x2+4x+5⇒x2+4x−5=0.10=x^2+4x+5\Rightarrow x^2+4x-5=0.
  2. Factor: (x+5)(x−1)=0⇒x=1(x+5)(x-1)=0\Rightarrow x=1 (reject x=−5x=-5). So x0=1x_0=1. …

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